We extend the functor to non-unital -algebras and make sure that the new extension agrees with the old definition given for unital -algebras.
Then we show that the functor is split exact, half-exact, homotopy invariant and stable under amplifications.
Since the functor is split exact, we have that no matter whether the algebra is unital or not, the short exact sequence which arises from the unitization, induces an exact sequence of the k-theory groups.
The map is when is unital, and if it is not unital, then it is the inclusion map.
The induced diagram has a commuting square on the right, since is functoriality for unital -algebras, and then in order to get a commuting square on the left, must be defined to be the restriction of to .
- for any -algebra.
- If and are -algebras, and if and then
- If are homotopic -homomorphisms, then
- If are homotopy equivalent -algebras, then is isomorphic to . More specifically, if is a homotopy, then are isomorphisms and .
4.2 The standard picture of the group K0
Proposition 4.2.2 and Lemma 4.2.3 below contain the standard picture of a handy description of what elements in look like. These results will be invoked whenever explicit calculations involving - groups are performed.
Definition 4.2.1 (The Scalar mapping)
Consider the split exact sequence obtained by adjoining a unit to the -algebra . Define the scalar mapping to be In other words, for all and . Notice that , and that belongs to for each . Let be the star homomorphism induced by . The image of is the subset of consisting of all matrices with scalar entries, and again for all elements in the unitzation. An element in or will be called a scalar element if .
The scalar mapping is natural in the sense that if and are -algebras, and if is a star homomorphism, then we get a commuting diagram, where .
Proposition 4.2.2. (The standard picture of General case)
One has that for each -algebra , that Moreover, the following hold.
- For each pair of projections the following conditions are equivalent:
(a).
(b). there exist natural numbers such that , in other words, stable equivalence.
(c). there exist scalar projections and such that
- If satisfies , then there is a natural number with .
- If is a star homomorphism , then
for all projections in .
Lemma 4.2.2
Let and be -algebras, and let be a -homomorphism. Suppose that belongs to the kernel of .
-
There exist a natural number , a projection , and a unitary in such that and
In other words, if is in the kernel of the induced K0 map, then its form in the k theory of B is standard with a projection “p”, and the image of p under the unitized map is unitarily equivalent to its own scalar part
-
If is surjective, then there is a projection , , and . In other words, the unitary is just 1!.
4.3 Half and split exactness and stability of K0
We begin with a description of what happens when units are adjoined to a short exact sequence. The straightforward proof is left to us.
Lemma 4.3.1.
Let be a short exact sequence of -algebras, and let .
- The mapping is invective.
- An element a in belongs to the image of if and only if
Example 4.3.5
From the split exactness, and traces, we conclude that